A242861 Triangle T(n,k) by rows: number of ways k dominoes can be placed on an n X n chessboard, k>=0.

1, 1, 1, 4, 2, 1, 12, 44, 56, 18, 1, 24, 224, 1044, 2593, 3388, 2150, 552, 36, 1, 40, 686, 6632, 39979, 157000, 407620, 695848, 762180, 510752, 192672, 35104, 2180, 1, 60, 1622, 26172, 281514, 2135356, 11785382, 48145820, 146702793, 333518324, 562203148

Offset

0,4

Comments

Also, coefficients of the matching-generating polynomial of the n X n grid graph.

In the n-th row there are floor(n^2/2)+1 values.

Links

Alois P. Heinz, Rows n = 0..14, flattened

Ralf Stephan, Two dominoes on the 3x3 chessboard, illustration of T(3,2)=44.

Eric Weisstein's World of Mathematics, Grid Graph

Eric Weisstein's World of Mathematics, Matching-Generating Polynomial

Wikipedia, Matching polynomial

Index entries for sequences related to dominoes

Index entries for sequences related to matchings

Formula

T(n,1) = A046092(n-1), T(n,2) = A242856(n).

T(n,floor(n^2/2)) = A137308(n), T(2n,2n^2) = A004003(n).

sum(k>=0, T(n,k)) = A210662(n,n) = A028420(n).

T(n,3) = A243206(n), T(n,4) = A243215(n), T(n,5) = A243217(n), T(n,floor(n^2/4)) = A243221(n). - Alois P. Heinz, Jun 01 2014

Example

Triangle starts:
  1
  1
  1  4   2
  1 12  44   56    18
  1 24 224 1044  2593   3388   2150    552     36
  1 40 686 6632 39979 157000 407620 695848 762180 510752 192672 35104 2180
  ...

Maple

b:= proc(n, l) option remember; local k;
      if n=0 then 1
    elif min(l[])>0 then b(n-1, map(h->h-1, l))
    else for k while l[k]>0 do od; expand(`if`(n>1,
         x*b(n, subsop(k=2, l)), 0) +`if`(k<nops(l) and l[k+1]=0,
         x*b(n, subsop(k=1, k+1=1, l)), 0) +b(n, subsop(k=1, l)))
      fi
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$n])):
seq(T(n), n=0..8); # Alois P. Heinz, Jun 01 2014

Mathematica

b[n_, l_List] := b[n, l] =  Module[{k}, Which[n == 0, 1, Min[l]>0, b[n-1, l-1], True, For[k=1, l[[k]]>0, k++]; Expand[If[n>1, x*b[n, ReplacePart[l, k -> 2]], 0] + If[k<Length[l] && l[[k+1]] == 0, x*b[n, ReplacePart[l, {k -> 1, k + 1 -> 1}]], 0] + b[n, ReplacePart[l, k -> 1]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, Array[0&, n]]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jun 16 2015, after Alois P. Heinz *)

Prog

(Sage)
def T(n, k):
   G = Graph(graphs.Grid2dGraph(n, n))
   G.relabel()
   mu = G.matching_polynomial()
   return abs(mu[n^2-2*k])

Crossrefs

Cf. A046741, A096713.

Sequence in context: A174005 A152818 A302235 * A109244 A171650 A225476

Adjacent sequences: A242858 A242859 A242860 * A242862 A242863 A242864

Keyword

nonn,tabf

Author

Ralf Stephan, May 24 2014

Status

approved